Gluon Polarization Tensor in external field in SU(3) theory

New Physics and Quantim Chromodynamics at External Conditions 2011

May 5

Dnipropetrovsk

M. Khandoga, V. Skalozub

Introduction

• Magnetic field of order is spontaneously generated in QCD vacuum at high temperatureSuperdaisy resummations: A. O. Starinets, A. V. Vshivtsev, V. Ch. Zukovskii. Phys. Lett. B 322, 403 (1994)Lattice simulations: N.O. Agasian (2003), V. Demchik (2008).

• Cosmological proof: Fermi-LAT Collaboration found out, that relict intergalactic magnetic fields of order ~ 10-15 G are observed (Science, Vol. 328. no. 5979, pp. 725 – 729, April 2010).

• Peripheral collisions of heavy ions: magnetic field is generated by parts of nuclei, travelling by sides

4 2gB g T

SU(2)-gluodynamics in external field

2 21 1( )

4 2L F A D

ab acb cD A [ ] ( ) ( ) ( ) ( )a a a abc a aF A A x A x A x A x

[ [ ], [ ]]ab abc cD A D A F

- QCD Lagrangian

, - Ghost fields

Background gauge

31 2( ) .a aB x x B

( ) ( ) ( ).a a aA x B x Q x

Field potential А(х) is divided into external field B(x) and quantum fluctuations Q(x):

External field is chosen in the following form:

2 33 3 3

3 33

2

1 1 1( )

4 4 21 1

( )2 21 1

( )4 2

ab b ab b

bc c b bc ab b ab b c b

abc abe b c d e ab bgh

L F F D Q D Q F Q

gf F Q Q gf D Q D Q Q Q

g f f Q Q Q Q D Q L

Lagrangian in background gauge:

Ghost Lagrangian3 3 .ac ac a cD gf B

Charged basisSince external field is directed along 3rd axis in the color space, it is

convenient to introduce the following basis, which is called charged:

3Q Q )(2

1 21 iQQW

3

* 3

.

.

D igB

D igB

3 3 * 3 3

3 33

2 3 3 33

1 1 1 1( )( ) ( )( )

4 2

( ) ( )

1 1( 2 )

2 2 gh

L Q Q W W D W D W Q Q

ig F Q W W igQ W W W W

g à W W W W Q Q W W F Q L

3 3 3

* *

( )

( )

Q Q Q

W D W D W

W D W D W

Ã

Corresponding Feynman diagrams

3Q

W

W

3Q3Q W W

3Q

W

3

SU(3)-gluodynamics in external field

• Spatial structure remains unchanged• Now we have 8 degrees of freedom instead of 3

which results in 8 gauge particles. • One more external field is added, it has same

spatial orientation and directed along 8th axis in color space.

3 3 8 81 2( ) ( )a a aB x x B B

SU(3)-gluodynamics Lagrangian in background gauge

à 1 21

1( ),2

W Q Q

4 52

1( ),2

W Q Q

,Q Q Q

Let’s switch to charged basis:

1 21

1( ),2

W Q Q 6 7

3

1( ),2

W Q Q

1 3 ,D igB 32,3 8

3.

2 3

BD i g B

,Ã abcf - SU(3) group structure constants.

SU(3)-gluodynamics Lagrangian in charged basis

Neutral gluons sectorNeutral gluons do not interact with each other. We can write interaction

Lagrangians of both neutral gluons as a combination of SU(2)-like Lagrangians:

3 3 int(3 ) int(3 ) int(3 )free I II IIIL L L L L

8 8 int(8 ) int(8 )free II IIIL L L L Every interaction Lagrangian has a structure, identical to SU(2) case.

Thus the polarization operator of neutral gluons in SU(3) theory can be brought to SU(2) case, already researched by M.Bordag, V. Skalozub, Phys. Rev. D 75, 125003 (2007)

3 3 8 (2) 3 (2) 3 8

(2) 8 3

1 1 3( , , ) ( , ) ( , )

4 2 2

1 3 1( , ),

4 2 2

Q Neut SU Neut SU

Neut SU

k B B k B B k B B B

k B B B

8 3 8 (2) 3 8 (2) 8 33 1 3 3 3 1( , , ) ( , ) ( , ).

2 2 2 2 2 2Q Neut SU Neut SUk B B k B B B k B B B

In the recent paper (V.Skalozub, A. Strelchenko (2004)) it was found out, that two fields are generated. After reaching the deconfinement phase two fields are generated:

3 2 3 2

8 2

30,9989 .

2

g TB

Spontaneous generation of magnetic fields at high temperature

3 2

3 20,2976 ,

g TB

0 3 8( ) , ,dB T T T B B

But after reaching some temperature only one field remains: 0 dT T

3 22

0 3 82

1 1( ) (1 ) ; 0.

4 2

g TB T T B B

Hence the behavior of field-dependant quantities differs significantly at high temperature. Let’s illustrate it on Debye mass.

Debye mass

Sometimes it is convenient to use an inverted quantity:

If electrical potential is surrounded by plasma, it has a limited reach:

.)( Dr

r

er

QrV

DD

mr

1

In QFT Debye screening is caused by vacuum polarization. Debye mass can be obtained from polarization operator:

).0,0,,( 4442 kkBTmD

In finite-temperature QCD there is a well-known result:

.)63

( 222 TgNN

m fcD O. Kalashnikov (1984)

Debye mass of neutral gluons

)))(2

4775.0

2

8859.0(

2

3(

3

2 32222

)( 3 BT

iT

gTgmQD

)))(2

4775.0

2

8859.0(

2

3(

3

2 2222

)( 8 T

iT

gTgmQD

38

3

1

2

3BB 38

3

1

2

3BB

Debye mass slightly grows at high temperature:

8

2 42 2 2

0 2( )( ) (1 (0,8222 0,4431) 0,1289 )

3 (2 )dD Q

g gm T T T g T i

8

2 42 2 2

0 2( )( ) (1 (0,5799 0,3866) 0,3982 )

3 (2 )D Q

g gm T T g T i

3

2 42 2 2

0 2( )( ) (1 (0,7574 0,4082) 0,0853 )

3 (2 )dD Q

g gm T T T g T i

3

2 42 2 2

0 2( )( ) (1 (0,3486 0,1638) 0,4289 )

3 (2 )D Q

g gm T T g T i

Vertex term of the Lagrangian3 3 3

1 1 1 1 1 1

8 3 8 32 2 2 2 2 2

8 3 8 33 3 3 3 3 3

( )

3 1 1( ) ( )( )

2 3 3

3 1 1( ) ( )( )

2 3 3

ig Q W W Q W W Q W W

i g Q Q W W Q Q W W W W

i g Q Q W W Q Q W W W W

1

2abc a b cgf Q Q Q

1 2 3 1 2 3

2 3 1 2 3 1

3 2 1 3 2 1

2(

2

)

ig W W W W W W

W W W W W W

W W W W W W

aW aW

bW

cWa b c

Charged gluons sectorIn SU(3) theory charged gluons do interact with each other:

int(3 ) int(8 ) int( ) ,WII WII free II II I II IIIL L L L L

int(3 ) int(8 ) int( ).WIII WIII free III III I II IIIL L L L L

SU(2) case was researched in paper by M. Bordag and V. Skalozub Phys. Rev. D 77, 105013 (2008)

For polarization operators of charged gluons we get

int(3 ) int( ) ,WI WI free I I II IIIL L L L

3 8 (2) 3 8 (2) 3 8

int 3 8

3 1 3 1 1 3( , , ) ( , ( )) ( , ( ))

4 2 2 4 2 2

( , , ),

IIw Char SU Char SU

Char

p B B p B B B p B B B

p B B

3 8 (2) 3 int 3 8( , , ) ( , ) ( , , ),Iw Char SU Charp B B p B B p B B

3 8 (2) 3 8 (2) 3 8

int 3 8

1 1 3 3 1 3( , , ) ( , ( )) ( , ( ))

4 2 2 4 2 2

( , , ).

IIIw Char SU Char SU

Char

p B B p B B B p B B B

p B B

Charged gluons Debye mass

Expressions for Debye mass:

38

3

1

2

3BB

38

3

1

2

3BB

Dependence on temperature:

2 42 2 2

0 2( )( ) (1 (0,8419 0,4567) 0,0915 )

3 (2 )I dD W

g gm T T T g T i

2 2 2 2 3

( )

2 3 0.8859 0.4775( ( )( ))

3 2 2 2ID W

m g T g i BT T

2 2 2 2 3

( )

2 3 0.8859 0.4775( ( )( ))

3 2 2 2IID W

m g T g i BT T

2 2 2 2 3

( )

2 3 0.8859 0.4775( ( ))

3 2 2 2IIID W

m g T g i BT T

2 42 2 2

0 2( )( ) (1 (0,7128 0,3794) 0,0179 )

3 (2 )II dD W

g gm T T T g T i

2 42 2 2

0 2( )( ) (1 (0,8562 0,1794) 0,1128 )

3 (2 )III dD W

g gm T T T g T i

2 42 2 2

0 2( )( ) (1 (0,2965 0,1459) 0,0671 )

3 (2 )ID W

g gm T T g T i

2 42 2 2

0 2( )( ) (1 (0,4821 0,1786) 0,2465 )

3 (2 )IID W

g gm T T g T i

2 42 2 2

0 2( )( ) (1 (0,3579 0,1421) 0,5872 )

3 (2 )IID W

g gm T T g T i

Conclusions

• Gluon polarization operator in external field is obtained in SU(3) case. Significant differences with SU(2) gluodynamics are observed.

• The spontaneously generated external field appears to reduce Debye mass.

• Obtained result may be used for further research, finding gluon spectra and magnetic masses.

Thank you for attention!